If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3),

If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2)}, is R a partial order?

If R is a relation on the set {1, 2, 3} defined by R = {(1, 1), (2, 2), (3, 3), (1, 2)}, is R a partial order?

Step 1: Understand what a partial order is. A partial order is a relation that is reflexive, antisymmetric, and transitive.
Step 2: Check if R is reflexive. A relation is reflexive if every element is related to itself. In R, we have (1, 1), (2, 2), and (3, 3), which means R is reflexive.
Step 3: Check if R is antisymmetric. A relation is antisymmetric if for any (a, b) and (b, a) in R, a must equal b. In R, we do not have any pairs (a, b) and (b, a) where a is not equal to b, so R is antisymmetric.
Step 4: Check if R is transitive. A relation is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R. We have (1, 2) in R and (2, 2) in R, but (1, 2) is not implied by these pairs, so R is not transitive.
Step 5: Since R is not transitive, we conclude that R is not a partial order.

Partial Order – A relation that is reflexive, antisymmetric, and transitive.
Reflexivity – Every element is related to itself.
Antisymmetry – If (a, b) and (b, a) are in the relation, then a must equal b.
Transitivity – If (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.